Problem 42 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \((a b)^{2}\)

Step-by-Step Solution

Answer

Therefore, the simplified form of \((a b)^{2}\) is \(a^{2}*b^{2}\).

Step 1: Understand the given expression

Begin by understanding that \((a*b)^2\) means that the entire expression within the bracket is being squared. In other words, \(a*b\) must be multiplied by itself.

Step 2: Apply the exponent rule

In mathematics, the rule of exponents states that any term inside parentheses raised to an exponent can be expanded to each part inside the parentheses raised to that exponent. Hence, the given expression \((a b)^{2}\) can be rewritten as \(a^{2} * b^{2}\), which is a simplified form.

Step 3: Final Simplified Expression

Upon applying the exponent rule, the simplified expression becomes \(a^{2}*b^{2}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a fundamental concept in algebra, and it involves reducing an equation or expression to its simplest form. When you look at an expression like \((a b)^2\), simplifying it means finding a way to rewrite it that makes it easier to understand or work with. In this case, simplifying involves using the rules of exponents to "unpack" the expression inside the parenthesis.

• Firstly, understand that the expression \((a b)^2\) means \(a \times b\) squared, or \(a \times b \times a \times b\).
• This can be a bit cumbersome, so instead, you use exponent rules to simplify.
• The simplified form is \(a^2 \times b^2\), which maintains the same value but is easier to handle.

Using these strategies helps to cut down complex expressions into manageable pieces, making calculations more efficient and comprehensible.

Algebraic Expressions

Algebraic expressions involve combining numbers and variables with arithmetic operations like addition, subtraction, multiplication, and division. They often consist of terms that include variables (e.g., \(a, b\)) and constants (e.g., numbers like 2, 3). For the expression \((a b)^2\), the variables are \(a\) and \(b\).

• Each variable can stand for any number, and algebra is all about finding relationships between these variables.
• Expressions can be modified by applying rules, such as distributing exponents across the variables within parentheses.
• In our example, by separating \((a b)^2\) into \(a^2 \times b^2\), we are still discussing the same algebraic expression, just in a more accessible form.

Overall, understanding algebraic expressions is crucial for solving equations and finding unknown values.

Multiplication of Variables

Multiplying variables is an integral part of handling algebraic expressions. Variables can be thought of as placeholders that stand for numbers. When two variables are multiplied, as in \(a \times b\), they produce a new expression that represents a single value. In the context of exponents, such as \((a b)^2\), multiplying these variables became even more interesting.

• Each variable is being multiplied by itself, so \(a \times b \times a \times b\) can be rewritten using the power rule of exponents.
• This is expressed as \(a^2 \times b^2\), which states that each variable is squared individually before being multiplied together.
• The power rule helps not only simplify expressions, but also provides a systematic way to handle multiplication across variables.

Altogether, mastering multiplication and understanding how it applies to variables and exponents is key to solving more complex algebraic problems.